Undirected Determinant and Its Complexity
Abstract
We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.
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© The Author(s) 2023.
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Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 06 July 2023
Accepted: 04 June 2023
Revision received: 03 June 2023
Received: 08 November 2021
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